Puzzle for July 26, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
Add B to both sides of eq.5: D - B + B = A + E + F + B which becomes D = A + E + F + B In eq.3, replace D with A + E + F + B: B + C + F = A + E + F + B Subtract B and F from both sides of the above equation: B + C + F - B - F = A + E + F + B - B - F which simplifies to eq.3a) C = A + E
Hint #2
In eq.6, replace C with A + E (from eq.3a): F = (A + E - E) ÷ A which becomes F = (A) ÷ A which means F = 1
Hint #3
In eq.2, replace D with B + C + F (from eq.3): B + C + F + E = A + B + C Subtract B and C from both sides of the equation above: B + C + F + E - B - C = A + B + C - B - C which simplifies to F + E = A Substitute 1 for F: eq.2a) 1 + E = A
Hint #4
Substitute 1 + E for A (from eq.2a) in eq.3a: C = 1 + E + E which makes eq.3b) C = 1 + 2×E
Hint #5
Substitute 1 + E for A (from eq.2a), and 1 + 2×E for C (from eq.3b) in eq.4: 1 + E + 1 + 2×E = B which makes eq.4a) 2 + 3×E = B
Hint #6
Substitute 2 + 3×E for B (from eq.4a), 1 + 2×E for C (from eq.3b), and 1 for F in eq.3: 2 + 3×E + 1 + 2×E + 1 = D which makes eq.3c) 4 + 5×E = D
Solution
Substitute 1 + E for A (from eq.2a), 2 + 3×E for B (from eq.4a), 1 + 2×E for C (from eq.3b), 4 + 5×E for D (from eq.3c), and 1 for F in eq.1: 1 + E + 2 + 3×E + 1 + 2×E + 4 + 5×E + E + 1 = 21 which simplifies to 9 + 12×E = 21 Subtract 9 from both sides of the above equation: 9 + 12×E - 9 = 21 - 9 which makes 12×E = 12 Divide both sides by 12: 12×E ÷ 12 = 12 ÷ 12 which means E = 1 making A = 1 + E = 1 + 1 = 2 (from eq.2a) B = 2 + 3×E = 2 + 3×1 = 2 + 3 = 5 (from eq.4a) C = 1 + 2×E = 1 + 2×1 = 1 + 2 = 3 (from eq.3b) D = 4 + 5×E = 4 + 5×1 = 4 + 5 = 9 (from eq.3c) and ABCDEF = 253911